# Properties

 Degree $2$ Conductor $3528$ Sign $0.458 + 0.888i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + (−0.965 + 0.258i)3-s − 4-s + (0.258 − 0.448i)5-s + (0.258 + 0.965i)6-s + i·8-s + (0.866 − 0.499i)9-s + (−0.448 − 0.258i)10-s + (0.965 − 0.258i)12-s + (1.22 − 0.707i)13-s + (−0.133 + 0.5i)15-s + 16-s + (−0.499 − 0.866i)18-s + (−1.67 + 0.965i)19-s + (−0.258 + 0.448i)20-s + ⋯
 L(s)  = 1 − i·2-s + (−0.965 + 0.258i)3-s − 4-s + (0.258 − 0.448i)5-s + (0.258 + 0.965i)6-s + i·8-s + (0.866 − 0.499i)9-s + (−0.448 − 0.258i)10-s + (0.965 − 0.258i)12-s + (1.22 − 0.707i)13-s + (−0.133 + 0.5i)15-s + 16-s + (−0.499 − 0.866i)18-s + (−1.67 + 0.965i)19-s + (−0.258 + 0.448i)20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3528$$    =    $$2^{3} \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.458 + 0.888i$ Motivic weight: $$0$$ Character: $\chi_{3528} (2909, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3528,\ (\ :0),\ 0.458 + 0.888i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.8968971641$$ $$L(\frac12)$$ $$\approx$$ $$0.8968971641$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + (0.965 - 0.258i)T$$
7 $$1$$
good5 $$1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2}$$
11 $$1 + (-0.5 + 0.866i)T^{2}$$
13 $$1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}$$
17 $$1 + (0.5 + 0.866i)T^{2}$$
19 $$1 + (1.67 - 0.965i)T + (0.5 - 0.866i)T^{2}$$
23 $$1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (0.5 - 0.866i)T^{2}$$
41 $$1 + (0.5 - 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-0.5 - 0.866i)T^{2}$$
59 $$1 - 1.41T + T^{2}$$
61 $$1 - 1.93iT - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + iT - T^{2}$$
73 $$1 + (-0.5 - 0.866i)T^{2}$$
79 $$1 - 1.73T + T^{2}$$
83 $$1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2}$$
89 $$1 + (0.5 - 0.866i)T^{2}$$
97 $$1 + (-0.5 - 0.866i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$